The variety of nilpotent pairs $(A,B)$ with $[A,B] = λI$

TL;DR

This work studies the variety $X = \{(A,B) \in M_n(k)^2 : A,B \text{ nilpotent}, [A,B]=\lambda I, \lambda\in k\}$ over an algebraically closed field $k$ of characteristic $p>0$ with $p\mid n$, proving that when $n=pr$ the variety $X$ is irreducible of dimension $n^2$. The authors combine the representation theory of the first Weyl algebra in characteristic $p$ (via its center $Z(W)=k[x^p,y^p]$ and Morita theory) with Premet's irreducibility results for the commuting nilpotent pairs in $\mathfrak{gl}_n(k)$, employing a fiber-dimension analysis and a construction of 'good' pairs with $[A,B]=\lambda I$, $\lambda\neq0$. A further contribution is a new characteristic-zero-to-positive-characteristic argument showing that Premet's result in good characteristic follows from the characteristic-zero case, clarifying how many results in good characteristic descend from characteristic zero. Collectively, the paper deepens the understanding of nilpotent commuting varieties and connects them to Weyl-algebra representation theory and centralizer structure, with implications for related Hilbert schemes and Morita-equivalence techniques.

Abstract

Let $k$ be an algebraically closed field of characteristic $p >0$. We consider the variety of nilpotent pairs $(A,B)$ with $[A,B]=λI$, namely the set of pairs $ X = \{ (A,B) \in M_n(k) \times M_n(k) \mid A,B \text{ nilpotent}, [A,B]=λI, λ\in k \}$. We prove that if $n=pr$, then $X$ is irreducible of dimension $n^2$.

Theorems & Definitions (24)

• Theorem 1
• Proposition 1
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Remark 1
• Lemma 1